Spectroscopic (or colorimetric) method for pH measurement is a well-established technique, wherein a pH sensitive reagent (or dye) changes color based on the pH of the solution. The theory of spectroscopic pH measurement using a single reagent indicator is well known (see Bates, “Determination of pH: Theory and Practice” Chapter 6 (1964), incorporated by reference herein in its entirety) and can be characterized by the following equation:
                    pH        =                  pKa          +                      log            ⁢                                                  ⁢                                          γ                B                                            γ                A                                              +                      log            ⁢                                                  ⁢                          B              A                                                          (        1        )            where Ka is the thermodynamic acid dissociation constant for the reagent, which is a function of temperature and pressure; A, B are concentrations of the acid and base forms of the reagent, respectively; γA, γB are activity coefficients of the acid and base forms of the reagent, respectively, which are a function of temperature, pressure and ionic strength of the solution.
The ratio of the base form to the acid form (B/A) of the reagent indicator may be calculated from spectral measurements using the following equations:
                              C          T                =                  A          +          B                                    (        2        )                                          OD                      λ            ⁢                                                  ⁢            1                          =                                            ɛ              A                              λ                ⁢                                                                  ⁢                1                                      ⁢            lA                    +                                    ɛ              B                              λ                ⁢                                                                  ⁢                1                                      ⁢            lB                                              (        3        )                                          OD                      λ            ⁢                                                  ⁢            2                          =                                            ɛ              A                              λ                ⁢                                                                  ⁢                2                                      ⁢            lA                    +                                    ɛ              B                              λ                ⁢                                                                  ⁢                2                                      ⁢            lB                                              (        4        )                                          ODR                      λ            ⁢                                                  ⁢            1                                λ            ⁢                                                  ⁢            2                          =                              OD                          λ              ⁢                                                          ⁢              2                                            OD                          λ              ⁢                                                          ⁢              1                                                          (        5        )                                          B          A                =                                            ODR                              λ                ⁢                                                                  ⁢                1                                            λ                ⁢                                                                  ⁢                2                                                                    ɛ                B                                  λ                  ⁢                                                                          ⁢                  2                                            /                              ɛ                A                                  λ                  ⁢                                                                          ⁢                  1                                                              ⁢                                    (                              1                -                                                                            ɛ                      A                                              λ                        ⁢                                                                                                  ⁢                        2                                                                                    ɛ                      A                                              λ                        ⁢                                                                                                  ⁢                        1                                                                              ⁢                                      1                                          ODR                                              λ                        ⁢                                                                                                  ⁢                        1                                                                    λ                        ⁢                                                                                                  ⁢                        2                                                                                                        )                                      (                              1                -                                                                            ɛ                      B                                              λ                        ⁢                                                                                                  ⁢                        1                                                                                    ɛ                      B                                              λ                        ⁢                                                                                                  ⁢                        2                                                                              ⁢                                      ODR                                          λ                      ⁢                                                                                          ⁢                      1                                                              λ                      ⁢                                                                                          ⁢                      2                                                                                  )                                                          (        6        )            where ODλi is the optical density measured at wavelength λi; l is path length; A, B are the respective concentrations of acid and base forms of the reagent in the sample-reagent mixture; CT is the total reagent concentration in the sample-reagent mixture; εAλi, εBλi are molar extinction coefficients at wavelength λi for A, B, respectively; and ODRλ1λ2 is the optical density ratio as defined in Equation (5).
The pH of a sample can be determined using spectral measurements by substituting Equation (6) in Equation (1). Because the acid and base form concentrations appear only as a ratio in Equation (1) and the absolute concentration of the reagent does not appear in Equations (1) or (5), the pH calculation is independent of the reagent concentration and the volume of the reagent added to the sample. If the ionic strength of the sample is at least an order of magnitude greater than the reagent concentration in the reagent-sample mixture, then the ionic strength of the sample and the activity coefficients (see Equations (7) and (8) below) are independent of reagent concentration. Accordingly, the only requirements for reagent addition are that the reagent concentration be (a) within a range where Beer's law is satisfied, (b) below an upper limit depending on the buffer strength of the sample beyond which the addition of the reagent could alter the sample pH, and (c) high enough to allow a good signal-to-noise ratio.
The molar extinction coefficients for the acid and base forms are obtained by calibration using solutions having pH values wherein the reagent exists completely in either its acid form or its base form. Alternatively, if the total reagent concentration is known very accurately, pH may be determined by measuring absorption at a single wavelength. However, because small errors in absolute concentration (CT) can cause large errors in pH calculation, this method may not provide an accurate pH measurement. The pH of a sample may also be calculated using a continuous spectral scan in the relevant region and applying regression analysis to determine the base to acid ratio.
From Equation (1), as pH values move away from the pKa values of a given reagent, the acid or the base fraction of the reagent becomes very small. Due to the low signal-to-noise ratios, the error in the pH measurement increases as pH values move away from the pKa value. Thus, for example, for a pH value 2 units lower than the pKa value, the fraction of the base form of the reagent is only 1%. Consequently, the OD corresponding to the base form peak wavelength is very low, resulting in inadequate pH accuracy. With single reagent indicators, the typical range of pH measurement is limited to about 1 to 1.5 units on either side of the reagent's pKa value. Beyond this range, a different reagent with a more appropriate pKa value must be used. Accordingly, the spectroscopic technique is less flexible to implement because knowledge of the sample's pH range is required a priori so that the appropriate reagent indicator is selected.
A commonly used pH indicator uses a mixture of reagents to extend the range of pH measurement. Visual observation of color allows pH determination to within 1 unit (see Vogel, “Text-Book of Quantitative Inorganic Analysis” 3rd Edition, Chapter 1.30, page 59 (1961), incorporated by reference herein in its entirety). However, it is difficult to obtain highly accurate measurements using spectroscopic techniques because the visible spectrum of the mixed reagent is generally a cumulative addition of the spectra of individual reagents. Unless the individual spectra are well resolved, it is difficult to invert the fraction of each form for accurate pH calculation.
Additional factors must be considered when performing pH analysis downhole. For example, only a limited number of reagents (pH indicators) can be transported downhole, the nature of the sample cannot be determined a priori, and only limited spectral analysis can be performed downhole. Accordingly, it would be useful to have a broad pH indicator that allows for simple, yet accurate, pH determination. Further, it is difficult to control the amount of reagent added to the sample under investigation in the downhole environment. Accordingly, a pH measurement that is not dependent on reagent concentration would be preferred.
Accordingly, it is an object of the present invention to provide an indicator mixture that retains the advantages of the single reagent method and that is effective over a broad range of pH values.
It is a further object of the present invention to provide a spectroscopic technique to determine the pH of a sample with the accuracy levels comparable to single reagent spectroscopy over a broad range of pH values.
It is yet another object of the present invention to provide a reagent mixture suitable for use in a downhole environment.